Fernando Cordero

Common ancestor type distribution: A Moran model and its deterministic limit

In a Moran model with population size N, two types, mutation and selection, let hkN be the probability that the common ancestor is fit, given that the current number of fit individuals is k. First, we express hkN in terms of the tail probabilities of an appropriate random variable LN. Next, we show that, when N tends to infinity (without any rescaling of parameters or time), LN converges to a geometric random variable. We also obtain a formula for h(x), the limit of hkN when k/N tends to x∈(0,1). In a second step, we describe two ways of pruning the ancestral selection graph (ASG) leading to the notions of relevant ASG and of pruned lookdown ASG (pruned LD-ASG). We use these objects to provide graphical derivations of the aforementioned results. In particular, we show that LN is distributed as the asymptotic number of lines in the relevant ASG and as the stationary number of lines in the pruned LD-ASG. Finally, we construct an asymptotic version of the pruned LD-ASG providing a graphical interpretation of the function h.

Asymptotic proportion of arbitrage points in fractional binary markets

A fractional binary market is a binary model approximation for the fractional Black–Scholes model, which Sottinen constructed with the help of a Donsker-type theorem. In a binary market the non-arbitrage condition is expressed as a family of conditions on the nodes of a binary tree. We call “arbitrage points” the nodes which do not satisfy such a condition and “arbitrage paths” the paths which cross at least one arbitrage point. In this work, we provide an in-depth analysis of the asymptotic proportion of arbitrage points and arbitrage paths. Our results are obtained by studying an appropriate rescaled disturbed random walk.

Strong asymptotic arbitrage in the large fractional binary market

We study, from the perspective of large financial markets, the asymptotic arbitrage (AA) opportunities in a sequence of binary markets approximating the fractional Black–Scholes model. This approximating sequence was introduced by Sottinen and named fractional binary market. The large financial market under consideration does not satisfy the standard assumptions of the theory of AA. For this reason, we follow a constructive approach to show first that a strong AA (SAA) exists in the frictionless case. Indeed, with the help of an appropriate version of the law of large numbers and a stopping time procedure, we construct a sequence of self-financing trading strategies leading to the desired result. Next, we introduce, in each small market, proportional transaction costs, and we show that a slight modification of the previous trading strategies leads to a SAA when the transaction costs converge fast enough to 0.

A probabilistic view on the deterministic mutation–selection equation: dynamics, equilibria, and ancestry via individual lines of descent

We reconsider the deterministic haploid mutation–selection equation with two types. This is an ordinary differential equation that describes the type distribution (forward in time) in a population of infinite size. This paper establishes ancestral (random) structures inherent in this deterministic model. In a first step, we obtain a representation of the deterministic equation’s solution (and, in particular, of its equilibria) in terms of an ancestral process called the killed ancestral selection graph. This representation allows one to understand the bifurcations related to the error threshold phenomenon from a genealogical point of view. Next, we characterise the ancestral type distribution by means of the pruned lookdown ancestral selection graph and study its properties at equilibrium. We also provide an alternative characterisation in terms of a piecewise-deterministic Markov process. Throughout, emphasis is on the underlying dualities as well as on explicit results.

The First Passage Time of a Stable Process Conditioned to Not Overshoot

Consider a stable Lévy process

BINARY MARKETS UNDER TRANSACTION COSTS

X=(X_t,t\ge 0)

Asymptotic arbitrage in fractional mixed markets

X=(Xt,t≥0) and let